Trees and branches in Banach spaces
Authors:
E. Odell and Th. Schlumprecht
Journal:
Trans. Amer. Math. Soc. 354 (2002), 40854108
MSC (2000):
Primary 46B03
Published electronically:
May 20, 2002
MathSciNet review:
1926866
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: An infinite dimensional notion of asymptotic structure is considered. This notion is developed in terms of trees and branches on Banach spaces. Every countably infinite countably branching tree of a certain type on a space is presumed to have a branch with some property. It is shown that then can be embedded into a space with an FDD so that all normalized sequences in which are almost a skipped blocking of have that property. As an application of our work we prove that if is a separable reflexive Banach space and for some and every weakly null tree on the sphere of has a branch equivalent to the unit vector basis of , then for all , there exists a subspace of having finite codimension which embeds into the sum of finite dimensional spaces.
 [GS]
David
Gale and F.
M. Stewart, Infinite games with perfect information,
Contributions to the theory of games, vol. 2, Annals of Mathematics
Studies, no. 28, Princeton University Press, Princeton, N. J., 1953,
pp. 245–266. MR 0054922
(14,999b)
 [Ja1]
Robert
C. James, Uniformly nonsquare Banach spaces, Ann. of Math.
(2) 80 (1964), 542–550. MR 0173932
(30 #4139)
 [J2]
W.
B. Johnson, On quotients of 𝐿_{𝑝} which are
quotients of 𝑙_{𝑝}, Compositio Math.
34 (1977), no. 1, 69–89. MR 0454595
(56 #12844)
 [JZ]
W.
B. Johnson and M.
Zippin, Subspaces and quotient spaces of
(∑𝐺_{𝑛})_{𝑙_{𝑝}} and
(∑𝐺_{𝑛})_{𝑐₀}, Israel J. Math.
17 (1974), 50–55. MR 0358296
(50 #10762)
 [JRZ]
W.
B. Johnson, H.
P. Rosenthal, and M.
Zippin, On bases, finite dimensional decompositions and weaker
structures in Banach spaces, Israel J. Math. 9
(1971), 488–506. MR 0280983
(43 #6702)
 [K]
N.J. Kalton, On subspaces of c and extensions of operators into Cspaces, Q. J. Math. 52 (2001), 312328.
 [KW]
Nigel
J. Kalton and Dirk
Werner, Property (𝑀), 𝑀ideals, and almost
isometric structure of Banach spaces, J. Reine Angew. Math.
461 (1995), 137–178. MR 1324212
(96m:46022), http://dx.doi.org/10.1515/crll.1995.461.137
 [KOS]
H.
Knaust, E.
Odell, and Th.
Schlumprecht, On asymptotic structure, the Szlenk index and UKK
properties in Banach spaces, Positivity 3 (1999),
no. 2, 173–199. MR 1702641
(2001f:46011), http://dx.doi.org/10.1023/A:1009786603119
 [Ma]
Donald
A. Martin, Borel determinacy, Ann. of Math. (2)
102 (1975), no. 2, 363–371. MR 0403976
(53 #7785)
 [MMT]
B.
Maurey, V.
D. Milman, and N.
TomczakJaegermann, Asymptotic infinitedimensional theory of
Banach spaces, Geometric aspects of functional analysis (Israel,
1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser,
Basel, 1995, pp. 149–175. MR 1353458
(97g:46015)
 [Mi]
V.
D. Milman, Geometric theory of Banach spaces. II. Geometry of the
unit ball, Uspehi Mat. Nauk 26 (1971),
no. 6(162), 73–149 (Russian). MR 0420226
(54 #8240)
 [MT]
Vitali
D. Milman and Nicole
TomczakJaegermann, Asymptotic 𝑙_{𝑝} spaces and
bounded distortions, Banach spaces (Mérida, 1992) Contemp.
Math., vol. 144, Amer. Math. Soc., Providence, RI, 1993,
pp. 173–195. MR 1209460
(94m:46014), http://dx.doi.org/10.1090/conm/144/1209460
 [O]
E.
Odell, Applications of Ramsey theorems to Banach space theory,
Notes in Banach spaces, Univ. Texas Press, Austin, Tex., 1980,
pp. 379–404. MR 606226
(83g:46018)
 [W]
R. Wagner, Finite highorder games and an inductive approach towards Gowers' dichotomy, Annals of Pure and Applied Logic, 111 (2001), 3960.
 [Z]
M.
Zippin, Banach spaces with separable
duals, Trans. Amer. Math. Soc.
310 (1988), no. 1,
371–379. MR
965758 (90b:46028), http://dx.doi.org/10.1090/S00029947198809657580
 [GS]
 D. Gale and F.M. Stewart, Infinite games with perfect information, Contributions to the theory of games, Annals of Math. Studies, No.28, Princeton University Press (1953), 245266. MR 14:999b
 [Ja1]
 R.C. James, Uniformly nonsquare Banach spaces, Ann. of Math. (2) 80 (1964), 542550. MR 30:4139
 [J2]
 W.B. Johnson, On quotients of which are quotients of , Compositio Math. 34 (1977), 6989. MR 56:12844
 [JZ]
 W.B. Johnson and M. Zippin, Subspaces and quotient spaces of and , Israel J. Math. 17 (1974), 5055. MR 50:10762
 [JRZ]
 W.B. Johnson, H.P. Rosenthal and M. Zippin, On bases, finite dimensional decompositions and weaker structures in Banach spaces, Israel J. Math. 9 (1971), 488506. MR 43:6702
 [K]
 N.J. Kalton, On subspaces of c and extensions of operators into Cspaces, Q. J. Math. 52 (2001), 312328.
 [KW]
 N.J. Kalton and D. Werner, Property , ideals, and almost isometric structure of Banach spaces, J. Reine und Angew. Math. 461 (1995), 137178. MR 96m:46022
 [KOS]
 H. Knaust, E. Odell, and Th. Schlumprecht, On asymptotic structure, the Szlenk index and UKK properties in Banach spaces, Positivity 3 (1999), 173199. MR 2001f:46011
 [Ma]
 D.A. Martin, Borel determinacy, Annals of Math. 102 (1975), 363371. MR 53:7785
 [MMT]
 B. Maurey, V.D. Milman and N. TomczakJaegermann, Asymptotic infinitedimensional theory of Banach spaces, Oper. Theory: Adv. Appl. 77 (1994), 149175. MR 97g:46015
 [Mi]
 V. Milman, Geometric theory of Banach spaces II, Geometry of the unit sphere, Russian Math. Survey 26 (1971), no. 6, 79163 (translation from Russian). MR 54:8240
 [MT]
 V.D. Milman and N. TomczakJaegermann, Asymptotic spaces and bounded distortions, Banach Spaces (Mérida, 1992; BorLuh Lin and W.B. Johnson, eds.), Contemp. Math. 144 (1993), 173195. MR 94m:46014
 [O]
 E. Odell, Applications of Ramsey theorems to Banach space theory, Notes in Banach spaces, ed. H.E. Lacey, Univ. of Texas Press, Austin, TX (1980), 379404. MR 83g:46018
 [W]
 R. Wagner, Finite highorder games and an inductive approach towards Gowers' dichotomy, Annals of Pure and Applied Logic, 111 (2001), 3960.
 [Z]
 M. Zippin, Banach spaces with separable duals, Trans. AMS, 310 (1988), 371379. MR 90b:46028
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
46B03
Retrieve articles in all journals
with MSC (2000):
46B03
Additional Information
E. Odell
Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712
Email:
odell@math.utexas.edu
Th. Schlumprecht
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 778433368
Email:
schlump@math.tamu.edu
DOI:
http://dx.doi.org/10.1090/S0002994702029847
PII:
S 00029947(02)029847
Received by editor(s):
October 10, 2000
Received by editor(s) in revised form:
November 7, 2001
Published electronically:
May 20, 2002
Additional Notes:
Research supported by NSF
Article copyright:
© Copyright 2002
American Mathematical Society
